Given an elliptic curve $E/\mathbb{Q}$ of analytic rank at most one, Kolyvagin proved that the Tate-Shafarevich group $\Sha(E/\mathbb{Q})$ is finite using his theory of Euler systems. In the higher rank cases, Kolyvagin gave a conjectural description of the structure of the Selmer group in terms of the Euler system. This conjecture has recently been proved by Wei Zhang for a large class of elliptic curves. We shall explain Kolyvagin's conjecture, discuss some of its remarkable consequences and overview the strategy of the proof. To put things in context, we shall begin with a brief survey on the current status of the BSD conjecture.

This is a note prepared for an Alcove Seminar talk at Harvard, Spring 2014. Our main references are [1] and [2].

TopThe BSD conjecture

Let $E$ be an elliptic curve over a number field $K$. Recall that the BSD conjecture relates the arithmetic invariants of $E/K$ with the an analytic object, the $L$-function $L(E/K,s)$ (see a lowbrow introduction).

Conjecture 1 (BSD)
  • $L(E/K,s)$ has analytic continuation over $\mathbb{C}$.
  • (Rank conjecture). Denote the analytic rank $r_\mathrm{an}(E/K)=\ord_{s=1} L(E/K,s)$ and the Mordell-Weil rank (or, algebraic rank) $r_\mathrm{MW}(E/K)=\rank E(K)$. Then $$r_\mathrm{an}=r_\mathrm{MW}.$$
  • (BSD formula) Denote $r=r_\mathrm{an}=r_\mathrm{MW}$. Then 
\frac{L^{(r)}(E/K,1)}{r!}&=\Omega(E/K)\cdot |\Sha(E/K)|\cdot R(E/K)\\&:= \frac{\displaystyle\prod_{v\mid \infty}\int_{E(K_v)}|\omega|_v\prod_{v\nmid\infty}c_v}{|d_K|^{1/2}}\cdot|\Sha(E/K)|\cdot\frac{\det(\langle P_i,P_j\rangle_{i,j=1}^r)}{|E(K)_\mathrm{tor}|^2}.
Here $\omega$ is the Neron differential of $\mathcal{E}/\mathcal{O}_K$ (see the remark), $c_v=[\mathcal{E}(K_v):\mathcal{E}^0(K_v)]$ is the local Tamagawa number, $\Sha(E/K)$ is the Tate-Shafarevich group, $\{P_i\}$ is a basis of the free part of $E(K)$, $\langle\cdot,\cdot\rangle$ is the Neron-Tate height paring on $E(K)$ and $d_K$ is the discriminant of $K$.
Remark 1 $\Omega(E/K)$ is called the period and $R(E/K)$ is called the regulator. Compare the BSD formula with classical class number formula $$\res_{s=1}\zeta_K(s)=\frac{2^{r_1}(2\pi)^{r_2}}{|d_K|^{1/2}}\cdot h(K)\cdot\frac{R(K)}{w(K)}.$$
Remark 2 In general, the module of Neron differentials of $\mathcal{E}/\mathcal{O}_K$ is a projective (but not necessarily free) module over $\mathcal{O}_K$: so the global Neron differentials over $\mathcal{O}_K$ may not exists. A more elegant way to define the period is then to use adelic integration (with convergence factors). But since we will mainly talk about elliptic curves over $\mathbb{Q}$ (and its base change over an imaginary quadratic field), we will not go into the details.
Remark 3 The BSD conjecture makes sense for general abelian varieties $A$ over number fields. The change in the BSD formula is to replace $|d_K|^{1/2}$ with $|d_K|^{\dim A/2}$, $|E(K)_\mathrm{tor}|^2$ with $|A(K)_\mathrm{tor}||\hat A(K)_\mathrm{tor}|$ and the height pairing with $\langle\cdot,\cdot\rangle: A(K)\times\hat A(K)\rightarrow \mathbb{R}$.

There has been vast progress on the BSD conjecture in recent years. We give a very brief (highly incomplete) summary.

Analytic continuation. Due to the recent work of someone in the audience, we now know $L(E/K,s)$ has analytic continuation for any totally real field $K$ of degree $\le2$.

Rank. To make clear the rank part of the conjecture, we recall the $p^\infty$-descent exact sequence $$0\rightarrow E(K) \otimes \mathbb{Q}_p/\mathbb{Z}_p\rightarrow\Sel_{p^\infty}(E/K)\rightarrow \Sha(E/K)[p^\infty]\rightarrow0. $$ The $p^\infty$-Selmer group is a direct sum of several (denoted by $r_p(E/K)$) copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and a finite abelian $p$-group. Conjecturally, $\Sha(E/K)[p^\infty]$ is finite, and hence conjecturally $$r_{p}=r_\mathrm{MW}$$ for any $p$. Consider the following implications for $K=\mathbb{Q}$, $$\xymatrix@R=2cm@C=0cm{& r_\mathrm{MW}=0, |\Sha[p^\infty]|<\infty \ar@{<=>}[ld]_{1)}& \\ r_p=0 \ar@{=>}[rr]^{3)}  & & r_\mathrm{an}=0 \ar@{=>}[lu]_{2)}}.$$ The implication 1) is trivial. The implication 2) is due to Gross-Zagier and Kolyvagin (80s) for all $p$. The implication 3) is due to Skinner-Urban (2000s) for $p\ge3$ good ordinary with surjective mod $p$ representation $\bar\rho_{E,p}:G_\mathbb{Q}\rightarrow \Aut(E[p])$. Now consider the analogous statement for rank 1 case, $$\xymatrix@R=2cm@C=0cm{& r_\mathrm{MW}=1, |\Sha[p^\infty]|<\infty \ar@{=>}[ld]_{1)}& \\ r_p=1 \ar@{=>}[rr]^{3)}  & & r_\mathrm{an}=1 \ar@{=>}[lu]_{2)}}.$$ The implication 1) is trivial. The implication 2) is again due to Gross-Zagier and Kolyvagin. The implication 3) is due to Wei Zhang (2013) for $p\ge5$ good ordinary with surjective $\bar\rho_{E,p}$ under some mild ramification hypothesis.

Remark 4 Notice in both cases, the implications 3) and 2) provide a purely algebraic criterion for determining the rank of $E(\mathbb{Q})$: together with Bhargava's counting method for 5-Selmer groups, Bhargava-Skinner-Zhang recently proved that at least 66% of all elliptic curves over $\mathbb{Q}$ satisfy the rank part of the BSD conjecture!

BSD formula. In the $r=0$ case, the $p$-part of the BSD formula is known for $p\ge3$ (under similar hypothesis) due to the work of Kato (2000s) and Skinner-Urban on the Iwasawa main conjecture for elliptic curves. In the $r=1$ case, the $p$-part of the BSD formula is proved by Wei Zhang for $p\ge5$ under similar hypothesis.

Remark 5 Notice nothing general is known for $p=2$! Unlike dealing with $p=2$ is not so popular in other fields of number theory, the $p=2$ case is in fact extremely interesting: the major terms contributing the BSD formula indeed come from mysterious powers of 2 and 3. Also, since quadratic twists preserve the 2-torsion structure, understanding the 2-part of the BSD formula is important for understanding the arithmetic of elliptic curves in quadratic twist families (e.g., the congruent number curve family $Dy^2=x^3-x$). In practice, the 2-descent is the easiest to carry out and provides a useful computational tool.

TopKolyvagin's Euler system

We now brief review the Euler system of Heegner points to motivate the statement of Kolyvagin's conjecture. In the next section, we shall state Wei Zhang's result on Kolyvagin's conjecture and deduce some of its consequences mentioned above.

Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$. Let $K/\mathbb{Q}$ be an imaginary quadratic extension with discriminant $d_K$ coprime to $N$ (we will assume $|d_K|>4$ for simplicity). Let $\eta_K: (\mathbb{Z}/d_K)^\times\rightarrow\{\pm1\}$ be the quadratic character associated to $K$. We assume the Heegner condition: every prime factor of $N$ splits in $K$. Then $\varepsilon(E/K)=\eta_K(-N)=-1$ and the BSD conjecture predicts that $r_\mathrm{MW}(E/K)$ is odd (in particular, $>0$). How do you construct a point?

The key thing is that under the Heegner condition, we have an abundant supply of algebraic points on $X_0(N)$ over the ring class fields of $K$. Recall that $X_0(N)$ classifies cyclic $N$-isogenies between two elliptic curves $E\rightarrow E'$. For $(n,N)=1$, let $\mathcal{O}_n=\mathbb{Z}+n\mathcal{O}_K$ be the order of $K$ of conductor $n$. Under the Heegner condition, there exists an ideal $\mathcal{N}\subseteq\mathcal{O}_K=\mathcal{O}_1$ with norm $N$. Then we have a pair of elliptic curves with CM by $\mathcal{O}_n$, $$\mathbb{C}/\mathcal{O}_n\rightarrow \mathbb{C}/(\mathcal{O}_n\cap\mathcal{N})^{-1}$$ with kernel $\cong \mathcal{O}_n/\mathcal{O}_n\cap\mathcal{N}\cong \mathbb{Z}/N \mathbb{Z}$. This defines a point (a Heegner point) $x_n\in X_0(N)$, which is defined over the ring class field $K_n$ of $K$ by the theory of complex multiplication. Here $K_n$ corresponds to the open compact subgroup $(\mathcal{O}_n\otimes \hat{\mathbb{Z}})^\times$ of $\mathbb{A}_K^\times$ under class field theory (so $K_1=H$ is the Hilbert class field of $K$). Using the modular parametrization $f: X_0(N)\rightarrow E$ (mapping $\infty$ to 0) we obtain algebraic points $y_n=f(x_n)\in E(K_n)$. Taking the trace using the group law on $E$, we construct a Heegner point $y_K=\tr_{K_1/K}y_1\in E(K)$.

Remark 6 The construction of a Heegner point depends on different choice of ideal classes (we used the trivial class $[\mathcal{O}_n]\in \Pic(\mathcal{O}_n)$ above, but any nontrivial class also works) and the choice of $\mathcal{N}$. Nevertheless it is well-defined up to sign and torsion, since the Atkin-Lehner involutions and $\Pic(\mathcal{O}_n)$ together act simply-transitively on all Heegner points; $[\infty-0]$ is torsion on $J_0(N)$.

The big theorem of Gross-Zagier we went through last semester is the following.

Theorem 1 (Gross-Zagier) $$L'(E/K,1)=\frac{\int_{E(\mathbb{C})}\omega\wedge\bar\omega}{|d_K|^{1/2}}\cdot\frac{1}{c^2}\cdot\langle y_K,y_K\rangle.$$ Here $c$ is the Manin constant defined so that $f^*\omega=c\cdot 2\pi i f_Edz$, here $f_E$ is the normalized eigenform associated to $E/\mathbb{Q}$.

It follows from the Gross-Zagier formula that $y_K$ is infinite order if and only if $r_\mathrm{an}(E/K)=1$. When $y_K$ is of infinite order, Kolyvagin used his theory of Euler system to prove the finiteness of $\Sha(E/K)$. A simple version of his theorem looks like

Theorem 2 (Kolyvagin) Suppose $y_K$ of infinite order. Let $p\ge3$ be prime such that $\bar\rho_{E,p}$ is surjective and $y_K$ is not divisible by $p$ in $E(K)$ (this is true for almost all $p$). Then $r_\mathrm{MW}(E/K)=1$ and $\Sha(E/K)[p]=0$.

The rough idea of Kolyvagin's proof is that he constructed a system of cohomology classes in $H^1(K,E[p])$ derived from the Heegner points $y_n$ satisfying nice norm and congruence relations. The local information about these explicit cohomology classes were good enough to bound the Selmer group $\Sel_p(E/K)$ (via the local and global Tate duality and the Chebotarev density). More precisely, let $\varepsilon=\varepsilon(E/\mathbb{Q})$ be the sign of the functional equation, then it was actually shown that $\Sel_p(E/K)^\varepsilon=0$ and $\Sel_p(E/K)^{-\varepsilon}=\mathbb{Z}/p \mathbb{Z}$ (the latter is contributed by $y_K$).

We now briefly recall the construction of Kolyvagin's Euler system. The key thing is the following assumption:

Definition 1 If a prime $\ell\nmid N p d_K$ is inert in $K$, and $$p\mid (\ell+1),\quad p\mid a_\ell,$$ where $a_\ell$ is the eigenvalue of the Hecke operator of $T_\ell$ on the modular from $f_E$. Then we say $\ell$ is a Kolyvagin prime. The positive integer $M(\ell)=\min\{v_p(\ell+1),v_p(a_\ell)\}$ is called the Kolyvagin index.
Remark 7 The condition $M(\ell)>0$ is equivalent to requiring $\Frob_\ell$ is in the conjugacy class of the complex conjugation in $\Gal(\mathbb{Q}(E[p])/\mathbb{Q})$. It follows from the Chebotarev density theorem that there is a positive density set of Kolyvagin primes.

The reason for this key condition is the following. Let $$G_\ell=\Gal(K_\ell/K_1)\cong \Pic(\mathcal{O}_\ell)/\Pic(\mathcal{O}).$$ Since $\ell$ is inert in $K$, we have $G_\ell\cong \mathbb{Z}/(\ell+1)=\langle\sigma\rangle$.

Definition 2 Define the Kolyvagin derivative operator $D_\ell=\sum_{i=1}^\ell i\sigma^i\in \mathbb{Z}[G_\ell]$ which satisfies the key identity $$(\sigma-1)D_\ell=\ell+1-\Tr_\ell.$$

Then $D_\ell y_\ell\in E(K_\ell)/p^M E(K_\ell)$ is indeed invariant under $\Gal(K_\ell/K_1)$ for $M\le M(\ell)$. To check this, we need to show that $(\sigma-1) D_\ell y_\ell$ lies in $p^ME(K_1)$. Namely $$(\ell+1-\Tr_\ell)(y_\ell)\in p^M E(K_1).$$ By assumption $p^M\mid\ell+1$, so it remains to show that $\Tr_\ell(y_\ell)\in p^ME(K_1)$. This is true since by assumption $p^M\mid a_\ell$: the trace of the Heegner point $y_{\ell}$ on $X_0(N)$ is exactly the Hecke operator $T_\ell$ acting on $y_1$, which projects to $a_\ell\cdot y_1\in E(K_1)$. The above relation is also the origin of the name Euler system: $\ell+1-a_\ell$ appears in the Euler factor of $L(E,s)$ at $\ell$.

More generally,

Definition 3 Let $n$ be a square-free product of Kolyvagin primes, then we say $n$ is a Kolyvagin number. The set of all Kolyvagin numbers is denoted by $\Lambda$. We denote $M(n)=\min_{\ell\mid n}M(\ell)$. Define $$D_n=\prod_{\ell\mid n}D_\ell\in \mathbb{Z}[G_n],$$ where $G_n=\Gal(K_n/K_1)\cong\prod_{\ell\mid n} G_\ell$. Then $D_ny_n\in H^1(K_n,E[p^M])$ is indeed invariant under the Galois group $\Gal(K_n/K_1)$ and descends to $H^1(K_1, E[p^M])$ as long as $M\le M(n)$ (because $E[p^M](K_n)=0$).
Definition 4 We define the Kolyvagin cohomology class $$c_M(n)=\sum_{s\in \Gal(K_1/K)}s D_ny_n\in H^1(K, E[p^M]),\quad M\le M(n).$$ The collection of cohomology classes $$\kappa=\{c_M(n)\in H^1(K,E[p^M]): n\in \Lambda, M\le M(n)\}$$ is called a Kolyvagin system.

TopKolyvagin's conjecture and consequences

Notice $c_M(1)$ is nothing but the cohomology class of $y_K$. In higher rank cases, the class $c_M(1)$ is indeed trivial by Gross-Zagier. Kolyvagin's Euler system constructs more classes $c_M(n)$ in $H^1(K,E[p^M])$. One can ask whether some of these classes could be nontrivial. An obvious but crucial observation is that this is the same as asking whether some $c_M(n)$ is not infinitely divisible by $p$.

Definition 5 For $r\ge0$, we define $\Lambda_r$ to be the set of all Kolyvagin numbers with exactly $r$ prime factors. Define $\mathcal{M}_r$ to be the $p$-divisibility of the Kolyvagin cohomology classes $c_M(n)$, $n\in \Lambda_r$. Namely, $\mathcal{M}_r$ is the largest integer $\mathcal{M}$ such that $p^{\mathcal{M}}\mid c_M(n)$ for all $n\in\Lambda_r$ and $M\le M(n)$.
Remark 8 Notice $\mathcal{M}_0$ is simply the $p$-divisibility of $y_K$. In particular, $\mathcal{M}_0<\infty$ if and only if $y_K$ is of infinite order.
Remark 9 One can describe the action of complex conjugation on these Kolyvagin classes $c_M(n)$ depending on $r$, the number of prime factors of $n$: $c_M(n)$ lies in the $\varepsilon_r=\varepsilon\cdot (-1)^{r+1}$-eigenspace (e.g., when $n=1$, $y_K$ lies in the $(-\varepsilon)$-eigenspace).

Kolyvagin proved that in fact $$\mathcal{M}_0\ge \mathcal{M}_1\ge\mathcal{M}_2\ge\cdots\ge0.$$ So increasing the number of prime factors of $n$ may help bring down the $p$-divisibility! We define $$\mathcal{M}_\infty=\lim_{r\rightarrow\infty} \mathcal{M}_r.$$

Remark 10 Clearly $\mathcal{M}_\infty<\infty$ if and only if $\kappa\ne\{0\}$.

The hope is that even in the case that $y_K$ is torsion, the cohomology classes $c_M(n)$ eventually becomes nontrivial.

Conjecture 2 (Kolyvagin) Suppose $p$ is an odd prime such that $\bar\rho_{E,p}$ is surjective. Then $\kappa\ne\{0\}$. Equivalently, $\mathcal{M}_\infty<\infty$.

Assuming this key conjecture, Kolyvagin's work allows us to understand the refined structure of $\Sel_{p^\infty}(E/K)$.

Definition 6 The vanishing order of $\kappa$ is defined to be $$\ord\kappa=\min\{\nu(n): c_M(n)\ne0, n\in \Lambda, M\le M(n)\}.$$ This is the smallest $r$ such that $\mathcal{M}_r<\infty$.
Remark 11 Clearly $\ord\kappa=0$ if and only if $y_k$ is of infinite order; Kolyvagin's conjecture is equivalent to $\ord\kappa<\infty$.

Starting from a nontrivial element by assumption $\nu=\ord\kappa<\infty$, Kolyvagin constructed the whole $p^\infty$-Selmer group (even in the higher rank case!). Moreover, its $\varepsilon_\nu$-eigenspace can be completely determined.

Theorem 3 (Kolyvagin) Suppose $\kappa\ne\{0\}$. Let $\nu=\ord\kappa<\infty$ (so $\mathcal{M}_v<\infty$).
  • $\Sel_{p^\infty}(E/K)$ is contained in the subgroup of $H^1(K,E[p^\infty])$ generated by $\kappa$.
  • $r_p^{\varepsilon_\nu}(E/K)=\nu+1$; $r_p^{-\varepsilon_\nu}(E/K)\le\nu$ and has the same parity as $\nu$. Let $d=\nu-r_p^{-\varepsilon_\nu}$ be the deficiency (so $d$ is even).
  • Let $\widetilde\Sha(E/K)[p^\infty]$ be the finite part of $\Sel_{p^\infty}(E/K)$. Write $$\widetilde\Sha(E/K)[p^\infty]^{\varepsilon_\nu}=\left(\bigoplus_{i\ge1}\mathbb{Z}/p^{a_i}\right)^2,\quad \widetilde\Sha(E/K)[p^\infty]^{-\varepsilon_\nu}=\left(\bigoplus_{i\ge1} \mathbb{Z}/p^{b_i}\right)^2,$$ where $\{a_i\},\{b_i\}$ are non-increasing. Then $$a_1=\mathcal{M}_{\nu+1}-\mathcal{M}_{\nu+2},\quad a_2=\mathcal{M}_{\nu+3}-\mathcal{M}_{\nu+4},\ldots,$$ and $$b_{d+1}=\mathcal{M}_\nu-\mathcal{M}_{\nu+1},\quad b_{d+2}=\mathcal{M}_{\nu+2}-\mathcal{M}_{\nu+3},\ldots.$$ In particular, $\widetilde\Sha(E/K)\ge p^{2(\mathcal{M}_\nu-\mathcal{M}_\infty)}$, with equality if and only if $d=0$.
Corollary 1 Suppose $\nu=0$ (i.e., the Heegner point $y_K$ is of infinite order). Then $d=0$, hence $r_\mathrm{MW}(E/K)=1$ and $|\Sha(E/K)|=p^{2(\mathcal{M}_0-\mathcal{M}_\infty)}$.

Now we are ready to state Wei Zhang's theorem precisely and then deduce some remarkable consequences.

Theorem 4 (Wei Zhang) Suppose $p\ge 5$ is good ordinary. Suppose $\bar\rho_{E,p}$ is surjective and ramifies at any $\ell||N$. Then $\mathcal{M}_\infty=0$. In particular, Kolyvagin's conjecture is true.
Remark 12 There is a similar version of this theorem for Euler systems of Heegner points on Shimura curves. Both versions are vital in the argument.
Corollary 2 Under the assumption of Theorem 4. If $r_p(E/\mathbb{Q})=1$, then $r_\mathrm{an}(E/\mathbb{Q})=r_\mathrm{MW}(E/\mathbb{Q})=1$ and $\Sha(E/\mathbb{Q})$ is finite.
Proof One can choose an imaginary quadratic field $K$ satisfying the Heegner hypothesis and $L(E^K/\mathbb{Q},1)\ne0$ for the quadratic twist $E^K/\mathbb{Q}$. Then by Gross-Zagier and Kolyvagin, $r_\mathrm{MW}(E^K/\mathbb{Q})=0$ and $\Sha(E^K/\mathbb{Q})$ is finite. So $r_p(E^K/\mathbb{Q})=0$. Therefore $r_p(E/K)=1$. Now by Kolyvagin's Theorem 3, b), $\nu+1$ is equal to rank of the larger piece of $\Sel_p(E/K)^{\pm}$, which implies $\nu=0$. So the desired result follows from Corollary 1.
Remark 13 Using the general version of Theorem 4 for Shimura curves, one can relax the ramification assumption for this corollary: e.g., when $N$ is square-free, one only needs to assume that $\bar\rho_{E,p}$ is ramified at any $\ell||N$ such that $\ell \equiv\pm1\pmod{p}$.
Corollary 3 Under the assumption of Theorem 4. If $r_\mathrm{an}(E/\mathbb{Q})=1$. Then the $p$-part of the BSD formula is true for $E/\mathbb{Q}$.
Proof Since $$\frac{\langle y_K,y_K\rangle}{[E(K): \mathbb{Z} y_K]^2}=\frac{\langle P,P\rangle}{|E(K)_\mathrm{tor}|^2},$$ comparing with the Gross-Zagier formula we know that the BSD formula for $E/K$ is equivalent to $$\frac{1}{c^2}=\frac{\Sha(E/K)\prod_{v\nmid\infty} c_v}{[E(K):\mathbb{Z}y_K]^2}.$$ It is known in general that that the any prime factor of Manin constant $c$ must divide $2N$, so $c$ is coprime to $p$. Under the assumption $\bar\rho_{E,p}$ ramifies at $\ell||N$, each $c_v$ is coprime to $p$. So the $p$-part of the BSD formula is equivalent to $$\Sha(E/K)=[E(K): \mathbb{Z} y_K]^2$$ up to $p$-adic units. By Theorem 3 c), this is equivalent to $$2(\mathcal{M}_0-\mathcal{M}_\infty)=2\mathcal{M}_0.$$ Namely, $\mathcal{M}_\infty=0$. This is exactly Theorem 4.
Remark 14 From the above argument we see that the BSD formula in the rank 1 can be translated into a (naive-looking) question about the divisibility of (derived) Heegner points, thanks to the Gross-Zagier formula. We also see that the apparently stronger Theorem 4 is indeed not stronger: $M_\infty=0$ is precisely predicted by the BSD conjecture.

TopOverview of the proof

Wei Zhang's proof beautifully blends various ideas and ingredients:

  • "Deforming" the $p$-Selmer group $\Sel_p(E/K)$ via congruences. For suitable level raising primes $q$, Ribet and Diamond-Taylor's level raising theorems allow one to find a newform of level $Nq$ that is congruent to the original newform of level $N$ mod $p$. In other words, one can keep the same $H^1(K, E[p])$ and vary the $p$-Selmer groups while only changing one local condition at $q$ at a time.
  • When the level raising prime $q$ is chosen to be inert in $K$, the root number changes. Gross-Parson proves that the $p$-Selmer parity changes as predicted (where the ramification assumption and $p\ge5$ is needed). Together with the Chebotarev density argument, one is able to lower the $p$-Selmer rank via lever-raising. In the $\Sel_p(E/K)$ rank one case, Gross-Parson relates the level raising $p$-Selmer rank $\Sel_p(A/K)$ with the local $p$-divisibility of the generator of $\Sel_p(E/K)$ at $q$.
  • To actually show that $c(1)$ is the generator of $\Sel_p(E/K)$, one uses the Jochnowitz congruence, which relates the local $p$-divisibility of the Heegner point at $q$ with the so-called algebraic part $L^\mathrm{alg}(A,K)$ mod $p$. To establish the Jochnowitz congruence, one needs the multiplicity one of the Hecke module of supersingular points on $X_0(N)_{\mathbb{F}_q}$ localized at a non-Eisenstein ideal, provided by Mazur's principle; and Gross's explicit Waldsburger formula on the definite quaternion algebra ramified at $q$.
  • To compute $L^\mathrm{alg}(A/K)$, one uses Skinner-Urban's BSD formula in the rank 0 case (where the good ordinary assumption is needed at the moment, but may be removed later). Showing $L^\mathrm{alg}(A/K)$ is nonzero mod $p$ needs the comparison between the congruence ideal of $f$, the modular degree of $X_0(N)\rightarrow E$ and the component group of the Neron model of $A$ at $q$. This was established by Ribet and Takahashi.
  • One then reduces the higher rank cases to the rank 1 case by successively lever-raising and rank lowering. To recover the nonvanishing of Kolyvagin's system from the rank one case, one uses a "triangularization" of $p$-Selmer group and the following cohomological congruences between Heegner points: the localization at $q_1$ of $c(n)$ on $X_0(N)$ is congruent to the localization at $q_2$ of $c(n)$ on the Shimura curve $X_0(N,q_1q_2)$. This is a combination of Jochnowitz congruences and Bertolini-Darmon congruences (used in the their proof of the anti-cyclotomic Iwasawa main conjecture). The latter asserts that the localization at $q_2$ of Heegner points on $X_0(N,q_1q_2)$ indeed factors through the component group at $q_2$, hence can be linked to the reduction of Heegner points on the components of the Cerednik-Drinfeld fiber $X_0(N,q_1q_2)_{\mathbb{F}_{q_2}}$, which matches up with the Hecke module of supersingular points on the definite quaternion algebra ramified at $q_1$. The cohomological congruence is then established by the multiplicity one.


[1]Wei Zhang, Selmer groups and the divisibility of Heegner points, 2013,

[2]Gross, Benedict H., Kolyvagin's work on modular elliptic curves, $L$-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., 153 Cambridge Univ. Press, Cambridge, 1991, 235--256.